A156364 Triangle read by rows: t(n,m) = Sum_{i=0..n} (-1)^(m-i)*Eulerian1(n-i+1, m-i) *Stirling2(n+i+1, i+1), where Eulerian1 are the Eulerian numbers of the first kind (A173018).

1, 1, 2, 1, 3, 19, 1, 4, 41, 274, 1, 5, 26, 812, 5521, 1, 6, -370, 1000, 20490, 143828, 1, 7, -3023, -8607, 34062, 640356, 4607296, 1, 8, -16977, -97974, -192901, 1249164, 23929389, 175377146, 1, 9, -83108, -703130, -1227484, -8076692, 53594570, 1040938950, 7739288201

Offset

0,3

Comments

Row sums are: {1, 3, 23, 320, 6365, 164955, 5270092, 200247856, 8823731317, 442515406465, 24893699411935,...}.

This sequence results from a substitution of the Eulerian numbers for the binomial in Smiley's Corollary 5.

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

L. Smiley, Completion of a Rational Function Sequence of Carlitz, page 3.

Formula

t(n,m) = Sum_{i=0..n} (-1)^(m-i)*Eulerian1(n-i+1, m-i)*Stirling2(n+i+1, i+1), where Eulerian1(n,k) = Sum_{j=0..k+1} (-1)^j * binomial(n+1, j)*(k+1-j)^n or Eulerian1(n,k) = A173018(n,k).

Example

Triangle begins as:
1;
1, 2;
1, 3,     19;
1, 4,     41,    274;
1, 5,     26,    812,    5521;
1, 6,   -370,   1000,   20490,  143828;
1, 7,  -3023,  -8607,   34062,  640356,  4607296;
1, 8, -16977, -97974, -192901, 1249164, 23929389, 175377146;

Mathematica

Eulerian1[n_, k_]:= Sum[(-1)^j Binomial[n+1, j](k+1-j)^n, {j, 0, k+1}];
t[n_, m_]:= Sum[(-1)^(m-i)*Eulerian1[n-i+1, m-i]*StirlingS2[n+i+1, i+1], {i, 0, n}];
Table[t[n, m], {n, 0, 10}, {m, 0, n}]//Flatten

Prog

(PARI)
Eulerian1(n, k) = sum(j=0, k+1, (-1)^j*binomial(n+1, j)*(k+1-j)^n);
t(n, m) = sum(i=0, n, (-1)^(m-i)*Eulerian1(n-i+1, m-i)*stirling(n+i+1, i+1, 2));
for(n=0, 10, for(m=0, n, print1(t(n, m), ", "))) \\ G. C. Greubel, Feb 24 2019
(Magma) [[(&+[(-1)^(m-j)*StirlingSecond(n+j+1, j+1)*(&+[(-1)^k*Binomial(n-j+2, k)*(m-j+1-k)^(n-j+1): k in [0..m-j+1]]): j in [0..m]]): m in [0..n]]: n in [0..10]]; // G. C. Greubel, Feb 25 2019
(Sage) [[sum((-1)^(m-j)*stirling_number2(n+j+1, j+1)*sum( (-1)^k*binomial(n-j+2, k)*(m-j-k+1)^(n-j+1) for k in (0..m-j+1)) for j in (0..m)) for m in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 25 2019

Crossrefs

Cf. A048993 (Stirling2), A156139, A156363, A173018.

Sequence in context: A132950 A197190 A247482 * A106169 A319493 A340202

Adjacent sequences: A156361 A156362 A156363 * A156365 A156366 A156367

Keyword

sign,tabl

Author

Roger L. Bagula, Feb 08 2009

Status

approved